Showing posts with label School Mathematics from Higher Viewpoints. Show all posts
Showing posts with label School Mathematics from Higher Viewpoints. Show all posts

2023-07-02

2: Why Are Mirror Images Regarded to Be Congruent in Plane Geometry?

<The previous article in this series | The table of contents of this series |

Just a definition by a mathematician? But there should be some good reasons for adopting the definition.

Topics


About: junior high school mathematics

The table of contents of this article

Starting Context

Target Context


  • The reader will know some reasons why mirror images are regarded to be congruent in the plane geometry.

Orientation


There is an article on becoming a benefactor of humanity by being a conduit of truths

Main Body

1: Why Are Mirror Images Regarded to Be Congruent in the Plane Geometry?


Special-Student-7-Rebutter
 In the (2-dimensional) plane geometry, 2 figures one of which is a mirror image of the other seems to be regarded to be congruent, at least in the Japanese junior high school mathematics.


Special-Student-7-Hypothesizer
 The Japanese junior high school mathematics seems to be saying that 2 figures one of which can be "moved" to coincide with the other is regarded to be congruent.

Special-Student-7-Rebutter
 What does "move" exactly mean?

Special-Student-7-Hypothesizer
 It is the problem that "move" is not defined explicitly in the Japanese junior high school mathematics, although I understand that junior high school students in general are not expected to understand rigorous definitions.

Special-Student-7-Rebutter
 But saying "move" is meaningless if the term is not understood accurately.

Special-Student-7-Hypothesizer
 Intuitively speaking, a triangle is imagined like a physical object, which is moved in the space: as normal junior high school students cannot teleport objects, they will imagine continuously moving the object in the space, without deforming the object.


Special-Student-7-Rebutter
 I see 2 problems there.

1st, how can you be sure that the object is not being inadvertently deformed while you are moving the object?

Special-Student-7-Hypothesizer
 Well, the physical object is supposed to be very hard and be never deformed unintentionally by my just moving it.

Special-Student-7-Rebutter
 Is "very hard" a legitimate mathematical concept?

Special-Student-7-Hypothesizer
 "very hard" is a physical assumption that if the object is wooden or something, the object will not be deformed, at least much, by being moved without being intentionally tortured. ... I know that it is not a refined mathematical concept, but it seems to have been a historical motivation for thinking of 'congruence'.

Special-Student-7-Rebutter
 2nd, what does "the space" mean when you say "is moved in the space"? As we are talking about a plane, "the space" should be naturally construed to be the plane, I think. Then, a figure cannot be "moved" to coincide with its mirror image by moving the figure continuously in "the space".

Special-Student-7-Hypothesizer
 Apparently, the Japanese junior high school mathematics "moves" the figure in the ambient 3-dimensional space.


Special-Student-7-Rebutter
 I do not say that they cannot introduce the ambient 3-dimensional space, but it seems unwise: while we should have been able to concentrate on the 2-dimensional plane, why do they have to complicate the situation by introducing the extra dimension?.

Special-Student-7-Hypothesizer
 They seem to have cursorily introduced the ambient 3-dimensional space, because they are familiar with the 3-dimensional space. However, they will have to introduce the ambient 4-dimensional space for the 3-dimensional space geometry, if they are consistent with the strategy.

Special-Student-7-Rebutter
 Certainly, 3-dimensional figures can be "moved" in the ambient 4-dimensional space, but it seems to be against the purpose of being intuitive: why do they have to make students think of the hard-to-imagine 4-dimensional space while we are concerned only with the 3-dimensional space?

Special-Student-7-Hypothesizer
 Introducing the ambient space seems to be a strategy that is natural only for the plane geometry, and employing a strategy that lacks generality seems unwise.

2: It Seems More Natural to Not Call Mirror Images 'Congruent', Supposing That 'Congruent' Means Being Same-Shaped and Same-Sized


Special-Student-7-Hypothesizer
 In fact, supposing that 'congruent' means being same-shaped-and-same-sized, calling mirror figures 'same-shaped' seems unnatural, in the first place.

Special-Student-7-Rebutter
 Are there some grounds for that supposition?

Special-Student-7-Hypothesizer
 At least most general (not mathematical) dictionaries say that 'congruent' means being same-shaped-and-same-sized.

Special-Student-7-Rebutter
 Mirror images are not same-shaped in my vocabulary.

Special-Student-7-Hypothesizer
 If someone feels mirror images being same-shaped, the reason seems to be that he or she has imagined moving figures in the ambient higher-dimensional space.

Special-Student-7-Rebutter
 But most people do not naturally imagine moving 3-dimensional figures in the ambient 4-dimensional space, I guess.

3: Some Reasons Why We Should Think of Map Instead of "Move"


Special-Student-7-Hypothesizer
 In fact, introducing "move" seems not wise, while "move" is understood as 'continuously move', as is naturally so.

A reason is that a figure has to be taken out of the plane into the ambient 3-dimensional space, if mirroring wants to be regarded to be a kind of "move".

Special-Student-7-Rebutter
 As has been discussed above.

Special-Student-7-Hypothesizer
 Another reason is that we have to think of the entire continuous movement of the figure in order for us to talk in terms of "move", while we are interested only in the original figure and the terminal figure.

So, why do we not skip the unnecessary middle?

Special-Student-7-Rebutter
 If there is a reason, it seems to be that they are not freed from the intuitive mental picture of moving a physical object.

Special-Student-7-Hypothesizer
 But in order for us to guarantee that the figure is not deformed in being moved, we have to guarantee that the figure is not deformed at each position, and then, why do we not guarantee that the figure is not deformed just at the terminal position?

Special-Student-7-Rebutter
 I do not guess that there is a legitimate reason.

Special-Student-7-Hypothesizer
 So, let us think of 'map' instead of "move", where 'map' means mapping each point of the original figure to a point of the terminal figure, without bothering to move the point continuously in the space.

The merits are the reverse of the unwiseness of introducing "move": we do not need the ambient space and we do not need to think of the middle.

Any map is prevalently denoted like \(f: S_1 \rightarrow S_2\), where \(S_1\) and \(S_2\) and some sets, and \(S_1\) is called 'the domain of the map' and \(S_2\) is called 'the codomain of the map'. The map does not need to really map the domain to the whole codomain, and the really mapped part of the codomain is called 'the range of the map'; if the range equals the codomain, the map is called to be a surjection. If any different 2 elements of \(S_1\) are mapped to different elements, the map is called to be an injection. Any map that is a surjection and an injection is called a bijection.

4: 2 Figures Are Congruent iff 1 of Them Is the Image of the Other Under an Isometry


Special-Student-7-Hypothesizer
 In fact, mathematically speaking, 2 figures are congruent iff 1 of them is the image of the other under an isometry, where any isometry is any map that preserves the distance between any 2 points.

In the plane geometry, with the sets of 2 figures denoted as \(S'_1 \subseteq \mathbb{R}^2\) and \(S'_2 \subseteq \mathbb{R}^2\), \(S'_1\) and \(S'_2\) are congruent if and only if there is an isometry, \(f: \mathbb{R}^2 \rightarrow \mathbb{R}^2\), such that \(f (S'_1) = S'_2\).

Special-Student-7-Rebutter
 Are mirror images congruent by the definition?

Special-Student-7-Hypothesizer
 Yes: the lengths of any corresponding sides of any 2 mirror triangles are the same.

Note that "preserves the distance between any 2 points" means that with the distance between 2 points, \(p_1, p_2\), denoted as \(dist (p_1, p_2)\) and the image of \(p_i\) denoted as \(f (p_i)\), \(dist (f (p_1), f (p_2)) = dist (p_1, p_2)\).

Special-Student-7-Rebutter
 Is any angle automatically preserved?

Special-Student-7-Hypothesizer
 Yes. Although the definition of isometry talks only about lengths, any triangle, \(ABC\), is mapped to \(A'B'C'\), and the angle, \(\angle ABC\), equals the angle, \(\angle A'B'C'\), because the side length, \(CA\), equals the side length, \(C'A'\), in addition that \(AB = A'B'\) and \(BC = B'C'\).

5: Isometry Is Really Any Combination of Translation, Rotation, and Mirroring


Special-Student-7-Hypothesizer
 Any isometry is really any combination of translation, rotation, and mirroring.




Special-Student-7-Rebutter
 Well, is that obvious?

Special-Student-7-Hypothesizer
 Although we do not show any rigorous proof, it is intuitively obvious.

Any translation preserves distances.

Any rotation preserves distances.

Any mirroring preserves distances.

And any isometry has to be a combination of translation, rotation, and mirroring, because when a triangle, \(ABC\), is mapped to \(A'B'C'\) by the isometry, \(A'B'C'\) is same-shaped-and-same-sized with or a mirror image of \(ABC\) (what else can it be indeed?), and \(A'B'C'\) has to have been mapped from \(ABC\) by a combination of translation, rotation, and mirroring.

Special-Student-7-Rebutter
 We are supposing that if \(A'B'C'\) is same-shaped-and-same-sized, it has to be a combination of translation and rotation, and if \(A'B'C'\) is a mirror image, it has to be a combination of translation, rotation, and mirroring.

Special-Student-7-Hypothesizer
 Although we have not proved the supposition rigorously, it seems intuitively obvious.

6: But Anyway, Why Is Isometry Used in the Definition of 'Congruence'?


Special-Student-7-Rebutter
 I understand the definition of 'congruence' with isometry, but why "isometry"?

Special-Student-7-Hypothesizer
 That is the point on which we have begun this article. Is a definition, "2 figures are congruent iff 1 of them is the image of the other under a combination of translation and rotation.", not more natural?

Someone may say that it is just that a mathematician (in fact, Euclid, I guess) just adopted the definition with isometry, but I think that that mentality is not good: mathematics is not any discipline in which students have to blindly accept an unreasonable definition just because an authority made the definition; we can and should ask whether there are good reasons to adopt the definition.

Special-Student-7-Rebutter
 What are the good reasons for the definition of 'congruence' with isometry?

Special-Student-7-Hypothesizer
 Some typical concerns in the plane geometry is that the length of a line segment equals the length of another line segment and an angle equals another angle, and isometry is sufficient for addressing such concerns.

We are typically concerned with whether an angle \(\angle ABC\) equals another angle \(\angle A'B'C'\), and it does not really matter whether the triangle, \(A'B'C'\), is same-shaped-and-same-sized with or a mirror image of the triangle, \(ABC\).

Special-Student-7-Rebutter
 Although there may be some cases in which it matters, but it is more economical to distinguish between being same-shaped-and-same-sized and being a mirror image only in such rare cases.

Special-Student-7-Hypothesizer
 That is the good reason why we have the concept that is defined with isometry.

Special-Student-7-Rebutter
 But there is the issue that whether the word, "congruent", is appropriate to represent the concept.

Special-Student-7-Hypothesizer
 As is discussed above, supposing that the general meaning of 'congruent' is being same-shaped-and-same-sized, the mathematical "congruent" is against the general meaning or along the general meaning only by introducing the ambient space. Either way, I feel that that use of "congruent" somehow misleading.

Special-Student-7-Rebutter
 To summarize, 2 figures are said to be congruent iff 1 of them is the image of the other under an isometry, and the reason why isometry is used there is that isometry is along our typical concerns of whether 2 line segment lengths are equal and whether 2 angles are equal. Whether "congruent" is a desirable word is an issue, but an mirror image can be regarded to be same-shaped-and-same-sized by introducing the ambient space, although whether introducing such an extra space is wise is another issue.

7: Mirroring on a Plane or in a 3-Dimensional Space as a Rotation in the Ambient Space


Special-Student-7-Hypothesizer
 Although we do not really need the ambient space in order to define 'congruence', "One of the 2 figures can be moved in the ambient space to coincide with the other figure." seems to be a view of congruence, and let us see how any mirroring is a rotation in the ambient space.

A mirroring on a plane will be easy to imagine as a rotation in the ambient 3-dimensional space.

Let us suppose that there is a triangle, \(ABC\), on the \(0 \lt x\) half of the \(x-y\) plane, and \(ABC\) is mirrored with respect to the \(y\) axis, with the mirror image, \(A'B'C'\) on the \(x \lt 0\) half of the plane.

As one will be able to imagine, when \(ABC\) is rotated \(\pi\) around the \(y\)-axis in the ambient 3-dimensional \(x-y-z\) space, \(ABC\) coincides with \(A'B'C'\).


More specifically, when any point, \((x, y, z)\), is rotated \(\theta\) around the \(y\) axis, the image, \(x', y', z'\), is \(\begin{pmatrix} x' \\ y' \\ z' \end{pmatrix} = \begin{pmatrix} cos \theta & 0 & - sin \theta \\ 0 & 1 & 0 \\ sin \theta & 0 & cos \theta \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix}\), by a matrix notation. If someone is not familiar with matrix notations, what that means is nothing but \(x' = cos \theta x - sin \theta z, y' = y, z' = sin \theta x + cos \theta z\).

Special-Student-7-Rebutter
 \(y\) is not changed as we are rotating the point around the \(y\) axis.

Special-Student-7-Hypothesizer
 When \(z = 0\), which means that the point is on the \(x-y\) plane, and \(\theta = \pi\), \(x' = - x, y' = y, z' = 0\), which is indeed the mirroring.

Let us see a mirroring in a 3-dimensional space as a rotation in the ambient 4-dimensional space.

Let us suppose that there is a tetrahedron, \(ABCD\), in the \(0 \lt x\) half of the \(x-y-z\) space, and \(ABCD\) is mirrored with respect to the \(y-z\) plane, with the mirror image, \(A'B'C'D'\) in the \(x \lt 0\) half of the space.


Special-Student-7-Rebutter
 While the mirroring on the plane was done with respect to a line, the mirroring in the 3-dimensional space is done with respect to a plane.

Special-Student-7-Hypothesizer
 Yes. In fact, a mirror you use to look at your face is a plane, not a line, right?

Special-Student-7-Rebutter
 I have not used any line mirror before.

Special-Student-7-Hypothesizer
 More generally, a mirroring in a \(\mathbb{R}^n\) space is done with respect to a \(n - 1\)-dimensional hyperplane. In fact, the line on the plane is a \(1\)-dimensional hyperplane.

When \(ABCD\) is rotated \(\pi\) around the \(y-z\) plane in the ambient 4-dimensional \(x-y-z-w\) space, \(ABCD\) coincides with \(A'B'C'D'\).

Special-Student-7-Rebutter
 How can I imagine "rotated \(\pi\) around the \(y-z\) plane"?

Special-Student-7-Hypothesizer
 Let us think in this way: rotations around the \(y\)-axis in the \(x-y-z-w\) space are 3-dimensional rotations in the \(x-z-w\) space.

Special-Student-7-Rebutter
 Ah, as only the \(y\)-axis is fixed, it is free in the \(x-z-w\) space.

Special-Student-7-Hypothesizer
 Let us restrict the 3-dimensional rotations in the \(x-z-w\) space to only the rotations around the \(z\)-axis, which is what we are talking about.

Special-Student-7-Rebutter
 So, they are the rotations with the \(y\)-axis and the \(z\)-axis fixed.

Special-Student-7-Hypothesizer
 And such any rotation can be parameterized by a single angle, \(\theta\), which is the angle of the 3-dimensional rotation in the \(x-z-w\) space around the \(z\)-axis.

When any point, \((x, y, z, w)\), is rotated \(\theta\) around that way, the image, \(x', y', z', w'\), is \(\begin{pmatrix} x' \\ y' \\ z' \\ w' \end{pmatrix} = \begin{pmatrix} cos \theta & 0 & 0 & - sin \theta \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ sin \theta & 0 & 0 & cos \theta \end{pmatrix} \begin{pmatrix} x \\ y \\ z \\ w \end{pmatrix}\), by a matrix notation. If someone is not familiar with matrix notations, what that means is nothing but \(x' = cos \theta x - sin \theta w, y' = y, z' = z, w' = sin \theta x + cos \theta w\).

Special-Student-7-Rebutter
 It has to be that because it is the 3-dimensional rotation in the \(x-z-w\) space around the \(z\)-axis while the \(y\)-axis is already fixed.

\(y\) and \(z\) are not changed of course.

Special-Student-7-Hypothesizer
 When \(w = 0\), which means that the point is in the \(x-y-z\) space, and \(\theta = \pi\), \(x' = - x, y' = y, z' = z, w' = 0\), which is indeed the mirroring.

Special-Student-7-Rebutter
 So, while most people probably cannot easily imagine rotations in the ambient 4-dimensional space, any mirroring in any 3-dimensional space is indeed a rotation in the ambient 4-dimensional space.

Special-Student-7-Hypothesizer
 A way to imagine the rotation in the 4-dimensional space is to think of the projections of the rotation into the \(x-y-z, x-y-w, x-z-w, y-z-w\) spaces.





For example, the projection of \((x, y, z, w)\) into the \(x-y-w\) space is \((x, y, w)\), and \(B: (1, 0, 0, 0)\) and \(D: (1, 0, 1, 0)\) map to the same \((1, 0, 0)\); the projection of \((x, y, z, w)\) into the \(y-z-w\) space is \((y, z, w)\), and \(B: (1, 0, 0, 0), C: (4, 0, 0, 0), B': (-1, 0, 0, 0), C: (-4, 0, 0, 0)\) map to the same \((0, 0, 0)\).

Special-Student-7-Rebutter
 We have seen the projections, so, what?

Special-Student-7-Hypothesizer
 Well, an important thing is that when you look at a 3-dimensional space, you are aware that the space is not the whole space but a projection, which means that each point is not really a point but a set of points with 1 parameter.

Special-Student-7-Rebutter
 So, when I extend my finger to touch a point in the projection, I have to be aware that I may not really touch the point in the 4-dimensional space, because my finger is touching the \(w = 0\) one in the set of points while the point is at \(w = 2\) for example.

Special-Student-7-Hypothesizer
 "Should be touching the point but not really touching the point" may seem supernatural but is not: the point is just at another \(w\).

Special-Student-7-Rebutter
 I see, ..., but so what?

Special-Student-7-Hypothesizer
 Well, beyond that, it is a matter of your imagination; in an analogue, you cannot really see any 3-dimensional object, either: you see 2-dimensional projections and your brain imagines the shape of the 3-dimensional object.

Special-Student-7-Rebutter
 So, it should be the same with a 4-dimensional object, you mean.

Special-Student-7-Hypothesizer
 I am guessing that it is a matter of practices that you begin to imagine the 4-dimensional object.

References
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2023-05-07

1: Why Is Negative Number Multiplied by Negative Number Plus?

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Just a definition by a mathematician? But there should be some good reasons for adopting the definition.

Topics


About: junior high school mathematics

The table of contents of this article

Starting Context

Target Context


  • The reader will know the reason why negative numbers are required, what integral numbers really are, the reasons for the arithmetic of integral numbers, and the reason for the order of integral numbers.

Orientation


There is an article on becoming a benefactor of humanity by being a conduit of truths

Main Body

1: Why Are Negative Numbers Required?


Special-Student-7-Hypothesizer
 Japanese students seem to learn negative numbers first at junior high school.

Special-Student-7-Rebutter
 Well, do Japanese elementary school students not know minus degrees Celsius?

Special-Student-7-Hypothesizer
 In Japan, temperatures are usually called in Celsius, and elementary school students in northern areas daily experience minus degrees Celsius in winters.

Special-Student-7-Rebutter
 But they do not know what minus degrees Celsius is ...

Special-Student-7-Hypothesizer
 At least officially.

In the ZFC set theory, the natural numbers set is first constructed, the integral numbers set is constructed from the natural numbers set, the rational numbers set is constructed from the integral numbers set, and the real numbers set is constructed from the rational numbers set.

In Japanese schools, positive rational numbers seem to be learned before integral numbers.

Special-Student-7-Rebutter
 I wonder why.

Special-Student-7-Hypothesizer
 That is an interesting point. Probably, the concept of negative number is supposed to include an abstraction that is rather harder to be grasped.

In fact, a 1 / 3 apple can be shown to students as a physical object, but how about a -1 apple?

Special-Student-7-Rebutter
 Students have to escape the notion that any number should correspond to some physical objects.

Special-Student-7-Hypothesizer
 We begin to need negative numbers when we begin to consider differences.

For example, an apple tree had 17 apples the last year and had 10 apples this year, then, what is the increase of the crops of the tree from the last year to this year?

Special-Student-7-Rebutter
 It is 7 decrease.

Special-Student-7-Hypothesizer
 I asked the "increase".

Special-Student-7-Rebutter
 I seem to be expected to say -7.

Special-Student-7-Hypothesizer
 Note that getting a difference is different from getting a remain.

For example, the apple tree had 10 apples, and 3 of them have been stolen by some crows, then, '10 - 3 = 7' apples remain; more than 10 apples cannot be stolen, and the remain cannot be negative.

The elementary school mathematics does not need to cope with negative numbers because subtraction in it is to get remains.

Special-Student-7-Rebutter
 The number 7 as a remain corresponds to the physical 7 remained apples, but the number -7 as a difference does not correspond to any specific physical objects but to the situation that the crops of the tree decreased in the year.

Special-Student-7-Hypothesizer
 Someone may say that that -7 corresponds to some 7 apples of the last year, but that is not true: in fact, which 7 apples does he or she mean?

Special-Student-7-Rebutter
 Probably, he or she picked up some arbitrary 7 apples, but -7 does not correspond to that specific 7 apples.

Special-Student-7-Hypothesizer
 We need negative numbers as decreases when increases are asked.

In the same vein, we need negative numbers as increases when decreases are asked.

In fact, the decrease of the crops of the tree from this year to the last year is -7: we can talk also about the difference from this year to the last year instead of from the last year to this year.

Special-Student-7-Rebutter
 I have seen an explanation that negative numbers are required in order to graduate a line.

Special-Student-7-Hypothesizer
 In fact, any line can be graduated without negative numbers: for example, mark an origin, mark 1, 3, 5, ... in one direction from the origin, and mark 2, 4, 6, ... in the other direction from the origin.

Special-Student-7-Rebutter
 In that way, certainly, the line can be graduated satisfying the requirement that each point is uniquely identified, but the expectation that the numbers should be monotonously increasing in one direction is not fulfilled.

Special-Student-7-Hypothesizer
 The line can in fact be graduated by only the open real numbers interval, \((0, \pi)\), by using \(tan^{-1} (x) + \pi / 2\), and also by any positive bounded open real numbers interval, (a, b), by scaling and translating \((0, \pi)\), then, it is monotonously increasing.

Special-Student-7-Rebutter
 A prevalent expectation is that the origin 0 should be at a finite location.

Special-Student-7-Hypothesizer
 Then, of course, negative numbers are required, because obviously, all the numbers in a direction from the origin have to be less than 0.

But anyway, the primal purpose of graduating the line should be to uniquely identify each point with a number, and for the purpose, negative numbers are not required.

2: Let Us Construct the Integral Numbers Set


Special-Student-7-Hypothesizer
 Let us think of the construction of the integral numbers set from the natural numbers set in this article, according to the arguments of the ZFC set theory, which means that we do not think of rational numbers.

An idea is to regard -7 to be the pair (10, 17), which means that -7 is the increase from 17 to 10.

Special-Student-7-Rebutter
 Is (17, 10) not more natural?

Special-Student-7-Hypothesizer
 Well, it is just the prevalent convention, which has been adopted probably because '10 - 17' is imagined.

Special-Student-7-Rebutter
 Anyway, also ones like (9, 16) should be valid.

Special-Student-7-Hypothesizer
 Yes, so, in fact, -7 = {(0, 7), (1, 8), ...}, which is an equivalent class in \(\mathbb{N} \times \mathbb{N}\) where \(\mathbb{N}\) is the natural numbers set.

The concept of 'equivalence class' may not be familiar to most junior high school students, but it is just a subset of a set when the set is divided into such subsets, where "divided into such subsets" means that any 2 different such subsets do not intersect and any element of the set is included in a subset.

Special-Student-7-Rebutter
 'equivalence class' is a concept you cannot escape in higher mathematics, because it appears often there.

In fact, we see many equivalence classes even in lower mathematics, like the even numbers subset and the odd numbers subset in the natural numbers set, although they are not called equivalence classes there.

Special-Student-7-Hypothesizer
 We need to define also positive integral numbers: 7 = {(7, 0), (8, 1), ...}.

Special-Student-7-Rebutter
 That may be a point that some junior high school students do not easily understand: "We already have the natural number 7; why do we need such another 7?".

Special-Student-7-Hypothesizer
 It is in fact an important point that integral 7 is really different from natural 7: natural 7 corresponds to 7 apples while integral 7 corresponds to the increase from 10 to 17, not to 7 apples.

Special-Student-7-Rebutter
 So, we need integral 7 because integral 7 and natural 7 are different in meanings.

Special-Student-7-Hypothesizer
 And we need it also in quest of symmetry: allowing (10, 17) but not (17, 10) is not symmetric. In other words, we want to treat negative integral numbers and positive integral numbers in the same way, so, we want negative integral numbers and positive integral numbers in the same shape.

Special-Student-7-Rebutter
 We will understand the convenience of it when we think of the arithmetic of integral numbers.

Special-Student-7-Hypothesizer
 {(0, 7), (1, 8), ...} is really denoted as [(0, 7)], because citing 1 element (0, 7) unambiguous determines the equivalence class. Instead, it can be [(1, 8)] or [(10, 17)] or something, because the purpose of identifying the equivalence class is fulfilled by any element of the equivalence class.

Special-Student-7-Rebutter
 So, we have some leeway of expressing -7 as [(0, 7)] or [(1, 8)] or something, but that is not any problem, because the equivalence class is unambiguously specified all right.

3: Some Notes


Special-Student-7-Hypothesizer
 In the following, just 7 is a natural number while [+7] or [-7] is an integral number.

The arithmetic of integral numbers is always done with only integral numbers.

So, '[+10] + [-7]' or '[+10] - [+7]' is allowed, but '[+10] - 7' is not allowed. You know, '[+10] - 7' is usually implicitly changed to '[+10] - [+7]', but this article does not allow such implicit changes, because such implicit changes hide exact mechanisms.

'[+10] + [-7]' and '[+10] - [+7]' are different things, although the results will turn out to be the same.

Although '[+10] - [+17]' has the answer, '10 - 17' has no answer, because the arithmetic of natural numbers has to have natural numbers as results. [-7] is [(10, 17)], not '10 - 17', which has no result.

Special-Student-7-Rebutter
 Of course, we are not saying that we need to always make such distinctions in daily lives: it is just that the purpose of this article requires such distinctions.

4: In the Axiomatic Theory, the Arithmetic Is Just Defined So, but


Special-Student-7-Hypothesizer
 In the following, we will examine why the arithmetic of integral numbers is as it is. For example, why [-1] + [-1] = [-2]?

Special-Student-7-Rebutter
 Someone may say that the arithmetic is just defines so by a mathematician.

Special-Student-7-Hypothesizer
 That stance is not logically wrong, but becomes a hindrance to deepen understandings, because any definition should have been made with some good reasons, and understanding the reasons clarifies the significance of the concept.

Any definition can be made up as far as it is well-defined, logically speaking, but we can and should ask why such a definition is wise.

Special-Student-7-Rebutter
 But how can a definition of the arithmetic of integral numbers be judged to be wise?

Special-Student-7-Hypothesizer
 A point is whether it accords with the interpretation of integral numbers as differences.

For example, what is the addition of 2 increases?

Special-Student-7-Rebutter
 When the 2 increases are the increase of the crops of the apple tree from 2 years ago to the last year and the increase from the last year to this year, the addition of the 2 will be the increase from 2 years ago to this year.

Special-Student-7-Hypothesizer
 Then, the addition of integral numbers should be wise to accord with that.

Special-Student-7-Rebutter
 That requirement seems to already uniquely determine the definition of the addition of integral numbers.

Special-Student-7-Hypothesizer
 Another point is whether it corresponds to the arithmetic of natural numbers when the concerned integral numbers are positive.

For example, we do not want '[+1] * [+2]' to be other than [+2], do we?

Special-Student-7-Rebutter
 Probably not: we want the arithmetic of integral numbers to be an extension of the arithmetic of natural numbers in a meaning.

Special-Student-7-Hypothesizer
 Another point is whether it satisfies the rules commonly expected to hold.

Special-Student-7-Rebutter
 What rules exactly?

Special-Student-7-Hypothesizer
 The commutativity, like '[-1] + [-2] = [-2] + [-1]', the associativity, like '[-1] + [-2] + [-3] = [-1] + ([-2] + [-3])', the distributability, like '[-1] * ([-2] + [-3]) = [-1] * [-2] + [-1] * [-3]', the reverse-ness of addition and subtraction, like '[0] - [-7] + [-7] = [0]', and the reverse-ness of multiplication and division, like '[+1] / [-7] * [-7] = [+1]', for example.

5: Addition of Integral Numbers


Special-Student-7-Rebutter
 What should '[-1] + [-1]' be?

Special-Student-7-Hypothesizer
 It is the accumulation of 1 decrease and 1 decrease, which should be 2 decrease, which means that '[-1] + [-1] = [-2]'.

Special-Student-7-Rebutter
 That seems the only reasonable option as far as we honor the interpretation of negative numbers as decreases and the interpretation of additions as accumulations.

Special-Student-7-Hypothesizer
 Also '[+1] + [+1] = [+2]' seems the only reasonable option.

Also '[-2] + [+1] = [-1]', '[-1] + [+2] = [+1]', '[+1] + [-2] = [-1]', '[+2] + [-1] = [+1]' come as only reasonable options.

As the exact definition, \([(n_{1, 1}, n_{1, 2})] + [(n_{2, 1}, n_{2, 2})] = [(n_{1, 1} + n_{2, 1}, n_{1, 2} + n_{2, 2})]\) will realize the above examples, where \(n_{i, j}\) is any natural number.

We can use that uniform definition for the entire integral numbers set because also positive integral numbers are formulated in the same shape with negative integral numbers.

Special-Student-7-Rebutter
 We need to check that that definition is a valid definition.

Special-Student-7-Hypothesizer
 That means that the result does not depend on the representatives of the equivalence classes. In fact, \([(n_{1, 1} + i, n_{1, 2} + i)] + [(n_{2, 1} + j, n_{2, 2} + j)] = [(n_{1, 1} + i + n_{2, 1} + j, n_{1, 2} + i + n_{2, 2} + j)] = [(n_{1, 1} + n_{2, 1}, n_{1, 2} + n_{2, 2})]\).

Special-Student-7-Rebutter
 Let us check whether the commutativity and the associativity hold.

Special-Student-7-Hypothesizer
 For the commutativity, \([(n_{1, 1}, n_{1, 2})] + [(n_{2, 1}, n_{2, 2})] = [(n_{1, 1} + n_{2, 1}, n_{1, 2} + n_{2, 2})] = [(n_{2, 1} + n_{1, 1}, n_{2, 2} + n_{1, 2})] = [(n_{2, 1}, n_{2, 2})] + [(n_{1, 1}, n_{1, 2})]\), because of the commutativity of natural numbers.

For the associativity, \([(n_{1, 1}, n_{1, 2})] + [(n_{2, 1}, n_{2, 2})] + [(n_{3, 1}, n_{3, 2})] = [(n_{1, 1} + n_{2, 1}, n_{1, 2} + n_{2, 2})] + [(n_{3, 1}, n_{3, 2})] = [(n_{1, 1} + n_{2, 1} + n_{3, 1}, n_{1, 2} + n_{2, 2} + n_{3, 2})] = [(n_{1, 1} + (n_{2, 1} + n_{3, 1}), n_{1, 2} + (n_{2, 2} + n_{3, 2}))] = [(n_{1, 1}, n_{1, 2})] + [(n_{2, 1} + n_{3, 1}, n_{2, 2} + n_{3, 2})] = [(n_{1, 1}, n_{1, 2})] + ([(n_{2, 1}, n_{2, 2})] + [(n_{3, 1}, n_{3, 2})])\), because of the associativity of natural numbers.

6: Subtraction of Integral Numbers


Special-Student-7-Rebutter
 What should '[-3] - [-7]' be?

Special-Student-7-Hypothesizer
 Let us think of '[-3] - [-7] + [-7]'.

Special-Student-7-Rebutter
 That is about the reverse-ness of addition and subtraction: subtracting any number and adding back the same number should mean no change.

Special-Student-7-Hypothesizer
 That means, '[-3] - [-7] + [-7] = [-3]'.

Then, the option for '[-3] - [-7]' is unique from the definition of addition of integral numbers. In fact, '\([(0, 3)] = [(n_1, n_2)] + [(0, 7)] = [(n_1, n_2 + 7)]\)' where \(n_i\) is a natural number, so, \((n_1, n_2) = (4, 0)\), for example; I mean, it can be also (5, 1) or something, but '[(4, 0)] = [(5, 1)]'.

As the exact definition, \([(n_{1, 1}, n_{1, 2})] - [(n_{2, 1}, n_{2, 2})] = [(n_{1, 1} + n_{2, 2}, n_{1, 2} + n_{2, 1})]\).

The definition is valid, because \([(n_{1, 1} + i, n_{1, 2} + i)] - [(n_{2, 1} + j, n_{2, 2} + j)] = [(n_{1, 1} + i + n_{2, 2} + j, n_{1, 2} + i + n_{2, 1} + j)] = [(n_{1, 1} + n_{2, 2}, n_{1, 2} + n_{2, 1})]\).

Special-Student-7-Rebutter
 The reason of that definition may not be immediately clear for someone.

Special-Student-7-Hypothesizer
 If \(0 \le n_{1, 1} - n_{2, 1}\) and \(0 \le n_{1, 2} - n_{2, 2}\), \([(n_{1, 1} - n_{2, 1}, n_{1, 2} - n_{2, 2})]\) will do, but that is not guaranteed, so, we have used \([(n_{1, 1}, n_{1, 2})] = [(n_{1, 1} + n_{2, 1} + n_{2, 2}, n_{1, 2} + n_{2, 1} + n_{2, 2})]\) in order to ensure that \(0 \le n_{1, 1} + n_{2, 1} + n_{2, 2} - n_{2, 1} = n_{1, 1} + n_{2, 2}\) and \(0 \le n_{1, 2} + n_{2, 1} + n_{2, 2} - n_{2, 2} = n_{1, 2} + n_{2, 1}\).

Special-Student-7-Rebutter
 We should check that that definition is consistent with the natural numbers arithmetic.

Special-Student-7-Hypothesizer
 When \(n_2 \le n_1\), \([+n_1] - [+n_2] = [(n_1, 0)] - [(n_2, 0)] = [(n_1, n_2)] = [(n_1 - n_2, 0)] = [+(n_1 - n_2)]\), which corresponds to \(n_1 - n_2 = n_1 - n_2\).

Special-Student-7-Rebutter
 Some important properties to be checked are \(i_1 - [+n_2] = i_1 + [-n_2]\) and \(i_1 - [-n_2] = i_1 + [+n_2]\) where \(i_1\) is any integral number and \(n_2\) is any natural number.

Special-Student-7-Hypothesizer
 \(i_1 - [+n_2] = [(n_{1, 1}, n_{1, 2})] - [(n_2, 0)] = [(n_{1, 1}, n_{1, 2} + n_2)] = [(n_{1, 1}, n_{1, 2})] + [(0, n_2)] = i_1 + [-n_2]\).

\(i_1 - [-n_2] = [(n_{1, 1}, n_{1, 2})] - [(0, n_2)] = [(n_{1, 1} + n_2, n_{1, 2})] = [(n_{1, 1}, n_{1, 2})] + [(n_2, 0)] = i_1 + [+n_2]\).

They are useful because now we can reduce any subtraction to an addition and the rules of additions can be employed.

For example, \(i_1 - [+n_2] - [+n_3] = i_1 + [-n_2] + [-n_3] = i_1 + [-n_3] + [-n_2] = i_1 - [+n_3] - [+n_2]\).

7: Multiplication of Integral Numbers


Special-Student-7-Rebutter
 What should '[-3] * [-7]' be?

Special-Student-7-Hypothesizer
 Let us think of '[-3] * [+7]' first.

It will be reasonable to assume that '[-3] * [+7] = [-3] + [-3] + [-3] + [-3] + [-3] + [-3] + [-3]'.

Special-Student-7-Rebutter
 That is the original concept of multiplication.

Special-Student-7-Hypothesizer
 So, '[-3] * [+7] = [-21]'.

Then, let us think of '[-3] * ([+7] + [-7])'.

What is it?

Special-Student-7-Rebutter
 '[+7] + [-7] = [0]' is already known, so, '[-3] * ([+7] + [-7]) = [-3] * [0]'.

Special-Student-7-Hypothesizer
 It is reasonably [0].

Now, let us suppose that the distribution law holds, which means that '[-3] * ([+7] + [-7]) = [-3] * [+7] + [-3] * [-7] = [0]'.

Then, as '[-3] * [+7] = [-21]', '[-3] * [-7]' has the unique option to be [+21].

Special-Student-7-Rebutter
 So, any negative number multiplied by any negative number is positive, which has come from making some natural requirements, among which is the distributability.

Special-Student-7-Hypothesizer
 The interpretation of '[-3] * [-7]' as [-7] times [-3] increases is probably enigmatic, but '[-3] * [-7]' has the good reason to be [+21].

As the exact definition, \([(n_{1, 1}, n_{1, 2})] * [(n_{2, 1}, n_{2, 2})] = [(n_{1, 1} * n_{2, 1} + n_{1, 2} * n_{2, 2}, n_{1, 1} * n_{2, 2} + n_{1, 2} * n_{2, 1})]\).

The motivation for that is '\((n_{1, 1} - n_{1, 2}) * (n_{2, 1} - n_{2, 2}) = n_{1, 1} * n_{2, 1} + n_{1, 2} * n_{2, 2} - (n_{1, 1} * n_{2, 2} + n_{1, 2} * n_{2, 1})\)', informally speaking.

Special-Student-7-Rebutter
 "informally speaking" means that that is not really allowed because '\(n_{1, 1} - n_{1, 2}\)' is not valid when '\(n_{1, 1} \lt n_{1, 2}\)': note that it is in the natural numbers arithmetic, not in the integral numbers arithmetic.

Special-Student-7-Hypothesizer
 It is just a motivation, not a valid calculation.

Anyway, the definition is valid, because \([(n_{1, 1} + i, n_{1, 2} + i)] * [(n_{2, 1} + j, n_{2, 2} + j)] = [((n_{1, 1} + i) * (n_{2, 1} + j) + (n_{1, 2} + i) * (n_{2, 2} + j), (n_{1, 1} + i) * (n_{2, 2} + j) + (n_{1, 2} + i) * (n_{2, 1} + j))] = [(n_{1, 1} * n_{2, 1} + i * n_{2, 1} + j * n_{1, 1} + i * j + n_{1, 2} * n_{2, 2} + i * n_{2, 2} + j * n_{1, 2} + i * j, n_{1, 1} * n_{2, 2} + i * n_{2, 2} + j * n_{1, 1} + i * j + n_{1, 2} * n_{2, 1} + i * n_{2, 1} + j * n_{1, 2} + i + j)] = [(n_{1, 1} * n_{2, 1} + n_{1, 2} * n_{2, 2}, n_{1, 1} * n_{2, 2} + n_{1, 2} * n_{2, 1})]\).

Special-Student-7-Rebutter
 Let us check the commutativity.

Special-Student-7-Hypothesizer
 '\([(n_{2, 1}, n_{2, 2})] * [(n_{1, 1}, n_{1, 2})] = [(n_{2, 1} * n_{1, 1} + n_{2, 2} * n_{1, 2}, n_{2, 1} * n_{1, 2} + n_{2, 2} * n_{1, 1})] = [(n_{1, 1} * n_{2, 1} + n_{1, 2} * n_{2, 2}, n_{1, 1} * n_{2, 2} + n_{1, 2} * n_{2, 1})] = [(n_{1, 1}, n_{1, 2})] * [(n_{2, 1}, n_{2, 2})]\)'.

Special-Student-7-Rebutter
 Let us check the associativity.

Special-Student-7-Hypothesizer
 '\([(n_{1, 1}, n_{1, 2})] * [(n_{2, 1}, n_{2, 2})] * [(n_{3, 1}, n_{3, 2})] = [(n_{1, 1} * n_{2, 1} + n_{1, 2} * n_{2, 2}, n_{1, 1} * n_{2, 2} + n_{1, 2} * n_{2, 1})] * [(n_{3, 1}, n_{3, 2})] = [((n_{1, 1} * n_{2, 1} + n_{1, 2} * n_{2, 2}) * n_{3, 1} + (n_{1, 1} * n_{2, 2} + n_{1, 2} * n_{2, 1}) * n_{3, 2}, (n_{1, 1} * n_{2, 1} + n_{1, 2} * n_{2, 2}) * n_{3, 2} + (n_{1, 1} * n_{2, 2} + n_{1, 2} * n_{2, 1}) * n_{3, 1})] = [(n_{1, 1} * (n_{2, 1} * n_{3, 1} + n_{2, 2} * n_{3, 2}) + n_{1, 2} * (n_{2, 1} * n_{3, 2} + n_{2, 2} * n_{3, 1}), n_{1, 1} * (n_{2, 1} * n_{3, 2} + n_{2, 2} * n_{3, 1}) + n_{1, 2} * (n_{2, 1} * n_{3, 1} + n_{2, 2} * n_{3, 2}))] = [(n_{1, 1}, n_{1, 2})] * [(n_{2, 1} * n_{3, 1} + n_{2, 2} * n_{3, 2}, n_{2, 1} * n_{3, 2} + n_{2, 2} * n_{3, 1})] = [(n_{1, 1}, n_{1, 2})] * ([(n_{2, 1}, n_{2, 2})] * [(n_{3, 1}, n_{3, 2})])\)'.

Special-Student-7-Rebutter
 Let us check the distributability.

Special-Student-7-Hypothesizer
 '\([(n_{1, 1}, n_{1, 2})] * ([(n_{2, 1}, n_{2, 2})] + [(n_{3, 1}, n_{3, 2})]) = [(n_{1, 1}, n_{1, 2})] * [(n_{2, 1} + n_{3, 1}, n_{2, 2} + n_{3, 2})] = [(n_{1, 1} * (n_{2, 1} + n_{3, 1}) + n_{1, 2} * (n_{2, 2} + n_{3, 2}), n_{1, 1} * (n_{2, 2} + n_{3, 2}) + n_{1, 2} * (n_{2, 1} + n_{3, 1}))] = [(n_{1, 1} * n_{2, 1} + n_{1, 2} * n_{2, 2} + n_{1, 1} * n_{3, 1} + n_{1, 2} * n_{3, 2}, n_{1, 1} * n_{2, 2} + n_{1, 2} * n_{2, 1} + n_{1, 1} * n_{3, 2} + n_{1, 2} * n_{3, 1})] = [(n_{1, 1} * n_{2, 1} + n_{1, 2} * n_{2, 2}, n_{1, 1} * n_{2, 2} + n_{1, 2} * n_{2, 1})] + [(n_{1, 1} * n_{3, 1} + n_{1, 2} * n_{3, 2}, n_{1, 1} * n_{3, 2} + n_{1, 2} * n_{3, 1})] = [(n_{1, 1}, n_{1, 2})] * [(n_{2, 1}, n_{2, 2})] + [(n_{1, 1}, n_{1, 2})] * [(n_{3, 1}, n_{3, 2})]\)'.

'\((i_2 + i_3) * i_1 = i_2 * i_1 + i_3 * i_1\)' where \(i_j\) is any integral number can be easily proved by using the commutativity, as '\((i_2 + i_3) * i_1 = i_1 * (i_2 + i_3) = i_1 * i_2 + i_1 * i_3 = i_2 * i_1 + i_3 * i_1\)'.

Special-Student-7-Rebutter
 An important property to be checked is \([-n_1] = [-1] * [+n_1]\) where \(n_1\) is any natural number.

Special-Student-7-Hypothesizer
 \([-n_1] = [(0, n_1)] = [(0, 1)] * [(n_1, 0)]\).

8: Division of Integral Numbers


Special-Student-7-Hypothesizer
 Note that as we are thinking of only integral numbers, we do not think about '[-7] / [-3]' or something.

Special-Student-7-Rebutter
 What should '[-21] / [+7]' be?

Special-Student-7-Hypothesizer
 We already have '[-3] * [+7] = [-21]'.

We suppose that '[-3] * [+7] / [+7] = [-3]', which is the reverse-ness of multiplication and division.

Then, '[-3] * [+7] / [+7] = [-21] / [+7] = [-3]'.

Special-Student-7-Rebutter
 What should '[-21] / [-7]' be?

Special-Student-7-Hypothesizer
 '[+7] * [-3] / [-3] = [+7]', by the reverse-ness of multiplication and division, while we already have '[+7] * [-3] = [-21]', so, '[+7] * [-3] / [-3] = [-21] / [-3] = [+7]'.

Special-Student-7-Rebutter
 What should '[+21] / [-3]' be?

Special-Student-7-Hypothesizer
 We already have '[-7] * [-3] = [+21]'. '[-7] * [-3] / [-3] = [+21] / [-3] = [-7]', by the reverse-ness of multiplication and division.

As the exact definition, '\([(n_{1, 1}, n_{1, 2})] / [(n_{2, 1}, n_{2, 2})]\)' is '\([((n_{1, 1} - n_{1, 2}) / (n_{2, 1} - n_{2, 2}), 0)]\)' when '\(n_{1, 2} \le n_{1, 1}\)' and '\(n_{2, 2} \lt n_{2, 1}\)'; '\([(0, (n_{1, 1} - n_{1, 2}) / (n_{2, 2} - n_{2, 1}))]\)' when '\(n_{1, 2} \le n_{1, 1}\)' and '\(n_{2, 1} \lt n_{2, 2}\)'; '\([(0, (n_{1, 2} - n_{1, 1}) / (n_{2, 1} - n_{2, 2}))]\)' when '\(n_{1, 1} \le n_{1, 2}\)' and '\(n_{2, 2} \lt n_{2, 1}\)'; '\([((n_{1, 2} - n_{1, 1}) / (n_{2, 2} - n_{2, 1}), 0)]\)' when '\(n_{1, 1} \le n_{1, 2}\)' and '\(n_{2, 1} \lt n_{2, 2}\)'.

Special-Student-7-Rebutter
 Does only the definition of division not look messy?

Special-Student-7-Hypothesizer
 If you are talking about having to make specifications in cases, it seems just a natter of appearance.

Special-Student-7-Rebutter
 Are appearances irrelevant?

Special-Student-7-Hypothesizer
 At least, appearances are deceptive.

Special-Student-7-Rebutter
 Anyway, an important property to be checked is \(i_1 / [-n_2] = [-1] * i_1 / [+n_2] = [-1] * (i_1 / [+n_2])\) where \(i_1\) is any integral number and \(n_2\) is any natural number.

Special-Student-7-Hypothesizer
 When \(i_1 = [+n_1]\), \(i_1 / [-n_2] = [(n_1, 0)] / [(0, n_2)] = [(0, n_1 / n_2)] = [-n_1] / [+n_2] = [-1] * [+n_1] / [+n_2] = [-1] * i_1 / [+n_2] = [-1] * [(n_1 / n_2, 0)] = [-1] * ([(n_1, 0)] / [(n_2, 0)]) = [-1] * (i_1 / [+n_2])\); when \(i_1 = [-n_1]\), \(i_1 / [-n_2] = [(0, n_1)] / [(0, n_2)] = [(n_1 / n_2, 0)] = [+n_1] / [+n_2] = [-1] * [-n_1] / [+n_2] = [-1] * i_1 / [+n_2] = [-1] * [(0, n_1 / n_2)] = [-1] * ([(0, n_1)] / [(n_2, 0)]) = [-1] * (i_1 / [+n_2])\).

Special-Student-7-Rebutter
 Let us check the commutativity.

Special-Student-7-Hypothesizer
 Before that, let us check that \(i_1 / i_2 / i_3 = i_1 / (i_2 * i_3)\), which makes the check of the commutativity easy.

Special-Student-7-Rebutter
 OK.

Special-Student-7-Hypothesizer
 When \(i_j = [+n_j]\), \(i_1 / i_2 / i_3 = [+n_1] / [+n_2] / [+n_3] = [(n_1, 0)] / [(n_2, 0)] / [(n_3, 0)] = [(n_1 / n_2, 0)] / [(n_3, 0)] = [(n_1 / n_2 / n_3, 0)] = [(n_1 / (n_2 * n_3), 0)] = [(n_1, 0)] / [(n_2 * n_3, 0)] = [(n_1, 0)] / ([(n_2, 0)] * [(n_3, 0)]) = [+n_1] / ([+n_2] * [+n_3]) = i_1 / (i_2 * i_3)\); when \(i_1 = [-n_1], i_2 = [+n_2], i_3 = [+i_3]\), \(i_1 / i_2 / i_3 = [-n_1] / [+n_2] / [+n_3] = [-1] * [+n_1] / [+n_2] / [+n_3] = [-1] * ([+n_1] / [+n_2]) / [+n_3]\), which is by the above property where \(i_1\) is taken to be \([+n_1]\), \(= [-1] * ([+n_1] / [+n_2] / [+n_3])\), which is by the above property where \(i_1\) is taken to be \([+n_1] / [+n_2]\), \(= [-1] * ([+n_1] / ([+n_2] * [+n_3])) = [-1] * [+n_1] / ([n_2] * [n_3])\), which is by the above property where \(i_1\) is taken to be \([+n_1]\), \(= [-n_1] / ([n_2] * [n_3]) = i_1 / (i_2 * i_3)\); when \(i_2 = [-n_2], i_3 = [+n_3]\), \(i_1 / i_2 / i_3 = i_1 / [-n_2] / [+n_3] = [-1] * (i_1 / [+n_2]) / [+n_3]\), which is by the above property where \(i_1\) is taken to be \(i_1\), \(= [-1] * (i_1 / [+n_2] / [+n_3])\), which is by the above property where \(i_1\) is taken to be \(i_1 / [+n_2]\), \(= [-1] * (i_1 / ([+n_2] * [+n_3])) = [-1] * (i_1 / [+(n_2*n_3)]) = i_1 / [-(n_2 * n_3)]\), which is by the above property where \(i_1\) is taken to be \(i_1\), \(= i_1 / ([-n_2] * [+n_3]) = i_1 / (i_2 * i_3)\); when \(i_2 = [+n_2], i_3 = [-n_3]]\), \(i_1 / i_2 / i_3 = i_1 / [+n_2] / [-n_3] = [-1] * (i_1 / [+n_2] / [+n_3])\), which is by the above property where \(i_1\) is taken to be \(i_1 / [+n_2]\), \(= [-1] * (i_1 / ([+n_2] * [+n_3])) = [-1] * (i_1 / [+(n_2*n_3)]) = i_1 / [-(n_2*n_3)]\), which is by the above property where \(i_1\) is taken to be \(i_1\), \(= i_1 / ([+n_2] * [-n_3]) = i_1 / (i_2 * i_3)\); when \(i_2 = [-n_2], i_3 = [-n_3]\), \(i_1 / i_2 / i_3 = i_1 / [-n_2] / [-n_3] = [-1] * (i_1 / [+n_2]) / [-n_3]\), which is by the above property where \(i_1\) is taken to be \(i_1\), \(= [-1] * [-1] * (i_1 / [+n_2]) / [+n_3]\), which is by the above property where \(i_1\) is taken to be \([-1] * (i_1 / [+n_2])\), \(= i_1 / [+n_2] / [+n_3] = i_1 / ([+n_2] * [+n_3]) = i_1 / ([-n_2] * [-n_3]) = i_1 / (i_2 * i_3)\).

Now, the proof of the commutativity is: \(i_1 / i_2 / i_3 = i_1 / (i_2 * i_3) = i_1 / (i_3 * i_2) = i_1 / i_3 / i_2\), because of the commutativity of multiplication.

9: Order of Integral Numbers


Special-Student-7-Rebutter
 What should the order of integral numbers be?

Special-Student-7-Hypothesizer
 That naturally comes from the interpretation of integral numbers as differences.

Which is the bigger increase between the [+7] increase and the [+3] increase?

Special-Student-7-Rebutter
 That should be about which case increased more.

Special-Student-7-Hypothesizer
 Then, '\([+3] \lt [+7]\)'.

Which is the bigger increase between the [+7] increase and the [-3] increase?

Special-Student-7-Rebutter
 The [-3] case did not increased at all.

Special-Student-7-Hypothesizer
 I think that '\([-3] \lt [+7]\)' is more reasonable.

Which is the bigger increase between the [-3] increase and the [-7] increase?

Special-Student-7-Rebutter
 [-7] is more far from increase.

Special-Student-7-Hypothesizer
 I think that '\([-7] \lt [-3]\)' is more reasonable.

References
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Table of Contents

1: Why Is Negative Number Multiplied by Negative Number Plus?
Just a definition by a mathematician? But there should be some good reasons for adopting the definition.
2: Why Are Mirror Images Regarded to Be Congruent in Plane Geometry?
Just a definition by a mathematician? But there should be some good reasons for adopting the definition.


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